Prove isomorphism

Swlabr

$$\displaystyle L\left(\begin{bmatrix}a&b\\c& d\end{bmatrix}\right)$$ $$\displaystyle \left(\begin{bmatrix}x\\y\\z\end{bmatrix}\right)$$ = 0?

I'm lost because I know it's not possible to multiply a 2x2 matrix by a 3x2 matrix
This isn't what you are trying to do. You want to find the kernel of the map L, which will be a 4-by-4 matrix but you don't want to think of it that way. You need to solve,

$$\displaystyle L(M)=0 \Rightarrow \ldots$$.

This is basically asking you if $$\displaystyle ax^3+bx^2+cx+d = 0$$ then what are $$\displaystyle a, b, c$$ and $$\displaystyle d$$?

Last edited:

Chris L T521

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This isn't what you are trying to do. You want to find the kernel of the map T, which will be a 4-by-4 matrix but you don't want to think of it that way. You need to solve,

$$\displaystyle T(M)=0 \Rightarrow \ldots$$.

This is basically asking you if $$\displaystyle ax^3+bx^2+cx+d = 0$$ then what are $$\displaystyle a, b, c$$ and $$\displaystyle d$$?
You mean 2x2 matrix...

wattkow

This isn't what you are trying to do. You want to find the kernel of the map T, which will be a 4-by-4 matrix but you don't want to think of it that way. You need to solve,

$$\displaystyle T(M)=0 \Rightarrow \ldots$$.

This is basically asking you if $$\displaystyle ax^3+bx^2+cx+d = 0$$ then what are $$\displaystyle a, b, c$$ and $$\displaystyle d$$?
Is it suppose to look like this:
$$\displaystyle L\left(\begin{bmatrix}a&b&0\\c&d&0\end{bmatrix}\right)$$ ?
I assume $$\displaystyle a, b, c, d = 0$$?

Swlabr

You mean 2x2 matrix...
No, 4-by-4. You are mapping from a 4-dimensional vector space into a 4-dimensional vector space...

The matrix you are mapping from is 2-by-2, and I was presuming the OP was getting confused by the linear map ~ matrix equivalence (it seemed like he was trying to find the kernel of the 2-by-2 matrix). That is why I wanted to point out that T is 4-by-4.

Swlabr

Is it suppose to look like this:
$$\displaystyle L\left(\begin{bmatrix}a&b&0\\c&d&0\end{bmatrix}\right)$$ ?
I assume $$\displaystyle a, b, c, d = 0$$?
Is what supposed to look like that?

(The map T in my earlier post is the same as the L in the rest of the thread. I have changed this to reflect that.)

wattkow

Is what supposed to look like that?
Bare with my as I am completely lost

I'm not sure how to find a kernel of an arbitrary matrix.

Swlabr

Bare with my as I am completely lost

I'm not sure how to find a kernel of an arbitrary matrix.
You aren't trying to find the kernel of an arbitrary matrix! (per se.)

Right, you have this linear map, denoted L, and you have said what it does. You now wish to find its kernel. That is, which matrices M are such that L(M)=0? That is, for which values of $$\displaystyle a, b, c, d$$ are such that $$\displaystyle ax^3+bx^2+cx+d = 0$$?

(I now have to go, sorry!)

wattkow

You aren't trying to find the kernel of an arbitrary matrix! (per se.)

Right, you have this linear map, denoted L, and you have said what it does. You now wish to find its kernel. That is, which matrices M are such that L(M)=0? That is, for which values of $$\displaystyle a, b, c, d$$ are such that $$\displaystyle ax^3+bx^2+cx+d = 0$$?

(I now have to go, sorry!)

The zero matrix?

Thanks for your help so far.

tonio

You aren't trying to find the kernel of an arbitrary matrix! (per se.)

Right, you have this linear map, denoted L, and you have said what it does. You now wish to find its kernel. That is, which matrices M are such that L(M)=0? That is, for which values of $$\displaystyle a, b, c, d$$ are such that $$\displaystyle ax^3+bx^2+cx+d = 0$$?

(I now have to go, sorry!)

With all due respect I think the OP is way more lost in this that what we could have guessed at the beginning, and I think he doesn't need help in this question but rather a thorough check over the whole material of this course, perhaps from the start...and the sooner the better, as things are going to get tougher pretty quick and without the elementary foundations is almost impossible to succeed.

I don't believe that a "thorough review" of all this stuff can be given within the frame of MHF but rather, imo, with the help of some private tutoring.

Tonio

wattkow

With all due respect I think the OP is way more lost in this that what we could have guessed at the beginning, and I think he doesn't need help in this question but rather a thorough check over the whole material of this course, perhaps from the start...and the sooner the better, as things are going to get tougher pretty quick and without the elementary foundations is almost impossible to succeed.

I don't believe that a "thorough review" of all this stuff can be given within the frame of MHF but rather, imo, with the help of some private tutoring.

Tonio
Yes, I'm in need of a thorough review but I don't have the luxury of time as my test is in an hour and my final is in a week. A simple example similar to the problem I listed would suffice